We derive an asymptotic theory of nonparametric estimation for a time series regression model $Z_t=f(X_t)+W_t$, where \ensuremath\{X_t\} and \ensuremath\{Z_t\} are observed nonstationary processes and $\{W_t\}$ is an unobserved stationary process. In econometrics, this can be interpreted as a nonlinear cointegration type relationship, but we believe that our results are of wider interest. The class of nonstationary processes allowed for $\{X_t\}$ is a subclass of the class of null recurrent Markov chains. This subclass contains random walk, unit root processes and nonlinear processes. We derive the asymptotics of a nonparametric estimate of f(x) under the assumption that $\{W_t\}$ is a Markov chain satisfying some mixing conditions. The finite-sample properties of $\hat{f}(x)$ are studied by means of simulation experiments.Comment: Published at http://dx.doi.org/10.1214/009053606000001181 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The classical nonstationary autoregressive models are both linear and Markov. They include unit root and cointegration models. A possible nonlinear extension is to relax the linearity and at the same time keep general properties such as nonstationarity and the Markov property. A null recurrent Markov chain is nonstationary, and β-null recurrence is of vital importance for statistical inference in nonstationary Markov models, such as, e.g., in nonparametric estimation in nonlinear cointegration within the Markov models. The standard random walk is an example of a null recurrent Markov chain.In this paper we suggest that the concept of null recurrence is an appropriate nonlinear generalization of the linear unit root concept and as such it may be a starting point for a nonlinear cointegration concept within the Markov framework. In fact, we establish the link between null recurrent processes and autoregressive unit root models. It turns out that null recurrence is closely related to the location of the roots of the characteristic polynomial of the state space matrix and the associated eigenvectors. Roughly speaking the process is β-null recurrent if one root is on the unit circle, null recurrent if two distinct roots are on the unit circle, whereas the others are inside the unit circle. It is transient if there are more than two roots on the unit circle. These results are closely connected to the random walk being null recurrent in one and two dimensions but transient in three dimensions. We also give an example of a process that by appropriate adjustments can be made β-null recurrent for any β ∈ (0, 1) and can also be made null recurrent without being β-null recurrent.

SAMANDRAG Vi diskuterer nokre grunnleggande tilnaermingar til eit datasett om normavvik («rettskrivingsfeil») i studenttekstar. Vi innfører modellar for å analysere tekstane, og vi ser på kva for type spørsmål datasettet kan svare på og kva det ikkje kan svare på. Då er det viktig å klargjere føresetnadane. Vi diskuterer òg forskjellen mellom å analysere eit datasett med tanke på å «bevise» samanhengar statistisk, og det å bruke datasettet for å fremje hypotesar. Statistiske metodar kan brukast i begge tilfella. Analysane her kan sjåast på som ei vurdering av kvaliteten i datasettet for ytterlegare analysar. NØKKELORD utforskande statistikk, normavvik i nynorsktekstar, modellar for teljedataABSTRACT We have a dataset about misspellings in Norwegian texts written by students. We introduce statistical models to analyze the dataset, and we discuss what questions can possibly be answered by the dataset. Moreover, we clearly point out the assumptions necessary for drawing conclusions. We also discuss different approaches to a dataset, like exploratory and confirmatory approaches. The analysis in this paper can be viewed as an evaluation of the quality of the data for further analysis.

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